The work of Kibble in [1]} is of particular interest. While many proposals for alternative non-linear `Schrödinger equations' focused on the non-relativistic regime [2]}[3]}, Kibble's work examined possible non-linear extensions for relativistic quantum dynamics. His approach was remarkably similar to our own — requiring a local Hamiltonian density to fulfill an algebraic relation analogous to the normal-normal Poisson bracket, eq.(). The treatment was not exhaustive, however; leaving it unclear what the full scope of these generalizations were. Our task in this section is thus to flesh out a wider range of potentials by explicitly solving the Poisson bracket relations for a desirable, but somewhat general, class of potentials.
